CCSSM – HS Session # 3
Functions
January 17, 2013
Functions are Fundamental
“Functions are truly fundamental to mathematics. In everyday
language we say, “The price of a ticket is a function of where you
sit,” or “The fuel needed to launch a rocket is a function of its
payload.” In each case, the word function expresses the idea that
knowledge of one fact tells us another. In mathematics, the most
important functions are those in which knowledge of one number
tells us another. If we know the length of the side of a square, its
area is determined. If the circumference of a circle is known, its
radius is determined.”
From Calculus: Single Variable, Second Edition, Hughes-Hallett,
Gleason, et al.
A Function is a Tool
“A mathematician needs functions for the same
reason that a builder needs hammers and drills.
Tools transform things. So do functions. In fact,
mathematicians often refer to them as
transformations because of this. But instead of
wood and steel, the materials that functions
pound away on are numbers and shapes, and,
sometimes, even other functions.”
Stephen Strogatz, The Joy of X
Formal Definitions of Functions
 Function Definition Activity
 In small groups, sort definitions
 Discuss with your group:
 What characteristics do the definitions
have in common?
 To what aspects of the function do all of
the definitions draw students’ and
teachers’ attention?
This activity and other content in this Powerpoint presentation is from the book
Developing Essential Understanding of Functions (9 – 12), NCTM (2010)
The Rule of Four
“Where appropriate, topics should be presented
 Geometrically
 Numerically
 Analytically
 Verbally
From Calculus: Single Variable, Second Edition, HughesHallett, Gleason, et al.
Outline for Functions Presentation
 Grade 8 Function Introduction
 HS Algebra I Function Topics
 Modeling Tasks
 Covariation and Rates of Change
 Sequences as Functions
 Growing Patterns as Functions
 Families of Functions
 Sample Tasks
HS CCSSM Functions Presentation
 HS Algebra II Function Topics
 Polynomial Functions
 Rational Functions
 Radical Functions
 Exponential Functions
 Logarithmic Functions as Solutions for
Exponentials
 Trigonometric Functions
 Sample Tasks
Function Introduction in Grade 8
 The Grade 8 CCSSM expects that students:
 Understand that a function is a rule that assigns to each input
exactly one output
 Understand that the graph of a function is a set of ordered pairs
consisting of an input and its corresponding output
 Are able to comfortable move between the different
representations in the Rule of Four
 Interpret y = mx+b as the equation of a linear function
 Be able to give examples of functions that are not linear
Function Introduction in Grade 8
 The Grade 8 CCSSM expects that students can
 Construct a function to model a linear relationship.
 Determine the rate of change and initial value,
given:
 A description of a the relationship
 Two (x,y) values (from a table or from a graph)
 Interpret the rate of change and the initial value
 In terms of the situation it models
 In terms of its graph or table of values
Function Introduction in Grade 8
 The Grade 8 CCSSM expects that students can
 Describe qualitatively the functional relationship
between two quantities by
 Analyzing a graph: increasing, decreasing,
linear, non-linear
Sketch a graph that
Exhibits given qualitative features of a
function or has been described verbally
CCSSM HS Functions
The Functions Standards are divided into four
broad categories, each with a set of clusters
Interpreting Functions (F-IF)
 Understand concept of and use notation
 Interpret functions that arise in terms of the
context
 Analyze functions using different
representations
CCSSM HS Functions
Building Functions (F-BF)
 Build a function that models a relationship
between two quantities
 Build new functions from existing function
CCSSM HS Functions
Linear, Quadratic & Exponential
Models (F-LE)
 Construct and compare linear, quadratic,
and exponential models and solve
problems.
 Interpret expressions for functions in terms
of the situation they model
CCSSM HS Functions
Trigonometric Functions (F-TF)
 Extend the domain of trigonometric
functions using the unit circle
 Model periodic phenomena with
trigonometric functions
 Prove and apply trigonometric identities
Interpreting Functions in Algebra I
 Understand the concept of a function and use
function notation
 Focus on
 Linear functions
 Exponential functions
 Quadratic functions
 Arithmetic sequences
 Geometric sequences
(F-IF.1, 2, 3) Read these standards
Interpreting Functions in Algebra I
 Graph functions expressed symbolically and show key
features by hand in simple cases, or by graphing
technology
 Analyze
 Linear functions (show intercepts)
 Exponential functions (show intercepts and end
behavior)
 Quadratic functions (show intercepts and
maxima/minima)
 Absolute value functions (show intercepts)
 Step functions
 Piecewise-defined functions
Building Functions in Algebra I
 Build a function that models a relationship between
two quantities (Read F.BF.1, 2)
 Linear
 Exponential
 Quadratic
 Build new functions from existing functions (Read
F.BF.3,4a)
 Linear only for F.BF.4a
 Quadratic
 Absolute Value
Models in Algebra I
 Construct and compare linear, quadratic, and
exponential models and solve problems
 Read F.LE.1a, 1b, 1c, 2, 3
 Interpret expressions for functions in terms of the
situation they model
Linear
Exponential of form f(x) = bx + k
Read F.LE.5
Circular Animal Pen
 Think about all possible circular animal pens
enclosed by a length of fencing. For each
length of fencing, there is the corresponding
area that the fencing will enclose when it is
formed into a circle.
 Describe the relationship between the area of
a circular animal pen and the length of the
fencing that it takes to enclose the pen.
From Developing Essential Understanding of Functions, NCTM (2010)
Circular Animal Pen
 When we use an equation such as A = L2/(2π)to describe
the relationship, we are implying the the “mapping”
definition. Each value of L maps to a value of A by
following the rule set forth in the equation.
 When we sketch of graph of the function, we view the
functions as consisting of ordered pairs of the form
[L, L2/(2π)].
 When we create a table, we bridge the mapping and
ordered pair definitions. Each entry in the input column
maps to the corresponding entry in the output column
AND the two entries in each row form an ordered pair.
Covariation and Rate of Change
 Using a covariation approach is an important first
step in developing a fuller comprehension of the
function concept.
 The rate of change of a function describes the
covariation between two variables (input and
output). In other words, how one variable quantity
changes with respect to the other variable.
 A function’s rate of change is one of the main
characteristics that determine what kinds of real
world phenomena the function can model.
Rate of Change
 Tables are a good way to observe covariation
From a covariation perspective (lens),
a function is understood as a
juxtaposition of two sequences,
each of which is generated
independently through a pattern
of values.
In the function at right, what stands out
from a covariation perspective?
x
f(x)
-1
1
0
1
1
3
2
7
3
13
4
21
Defining Rate of Change
 A rate of change is a rate!
change in output value
change in input value
 When students see only tables where the
change in input value is 1, they may think that
the rate of change is only the change in the
output value!
Defining Rate of Change
 Average and instantaneous rates of change
Suppose that it starts to rain at noon.
falls.
Between 2 PM and 6 PM a total of 12 millimeters of rain
What is the rate of change between noon and 6 PM?
What is the rate of change between 2 PM and 6 PM?
What is the rate of change between 3 PM and 4 PM?
What is the rate of change at 5 PM?
Looking at tables through
different perspectives
 When you look at the table from a
x
y
-2
4
-1
1
0
0
1
1
 When you look at the table from a
2
4
correspondence or mapping perspective,
3
9
4
16
covariation perspective,
what do you notice?
what you do notice?
Looking at Covariation Patterns
for Clues as to Function Family
 Research indicates that students typically use a covariation
perspective before attempting to generalize the relationships
with a mapping perspective
 The pattern of covariation gives students a clue as to what function
family the table describes
x
f(x) = 3x + 2
-1
-1
0
2
1
5
1
2
8
1
3
11
 A linear function has a constant rate of change
 The ratio of the change in output to the 1
change in input is constant
1
3
3
3
3
Looking at Covariation Patterns
for Clues as to Function Family
 A quadratic function has a linear rate of change
 As the input changes by one unit, the output changes at a constant
rate
 This constant rate can be seen by finding the second differences
x
f(x) = 3x2 + 2
-1
5
0
2
1
5
1
2
14
1
3
29
1
1
Linear Constant
-3
3
9
15
6
6
6
Rate of Change
 Exponential functions have a rate of change that is proportional to
the value of the function.
 Whenever the input is increased by 1 unit*, the output is
multiplied by a constant factor,
x
2x
which is the base.
Rate of change = 2*f(x)
*Note that this constant factor is
not as readily seen when the input
is increased by a value other than
1 unit.
-1 2-1 = ½
0
20 = ½ * 2 = 1
1
21
2
22
3
23 = 4*2 = 8
= 1*2 = 2
= 2*2 = 4
½
1
2
4
Sequences as Functions
Sequences represent a special type of function whose domain is
the counting numbers.
 Arithmetic sequences have a constant rate of change and are
related to linear functions.
 Geometric sequences have a variable rate of change and are
related to exponential functions.
 Growing patterns can have constant or variable rates of
change and are generally related to polynomial functions.
Sequences as Functions
 Arithmetic Sequences
A sequence is arithmetic if each term beyond the first term
can be obtained from the previous term by adding a
constant m (m can be positive, negative, or zero).
5, 9, 13, 17, 21…
1.6, 1.1, 0.6, 0.1, -0.4, -0.9, -1.4…
Recursive Rule: NEXT = NOW + m
Sequences as Functions
 Arithmetic Sequences
5, 9, 13, 17, 21…
1.6, 1.1, 0.6, 0.1, -0.4, -0.9, -1.4…
The Recursive Rule is easy to define: NEXT = NOW + m
What is the Recursive Rule of the above sequences?
Sequences as Functions
 Arithmetic Sequences
1, 4, 7, 10, 13…
and
1, 5, 9, 13, 17
The rule that assigns a sequence value to a specific counting,
such as “what is the 100th term in the sequence” is harder to
recognize.
Find the formulas in terms of N for the Nth entries of the
sequences. (Let N = 1 be the first entry in each series) Finding
the formula generally requires identifying what the the “0th”
entry in the sequence would be.
What relationship do you see between the recursive rules
(NEXT = NOW + m and the formulas (input the term # and
output the sequence value)?
Sequences as Functions
Geometric Sequences
 Define the recursive rule for each sequence below
 Find the formula in terms of N for any term in each
sequence
10, 20, 40, 80…
and
5, 5/3, 5/9, 5/27…
What relationship do you see between the recursive rules and
the formulas?
Problems related to F-IF.1, 2, 3
From Illustrative Mathematics:
Interpreting the Graph (Trig
function)http://www.illustrativemathematics.org/illustrations/
636
The Customers (Idea of Function in non-math
context)http://www.illustrativemathematics.org/illustrations/
624
Families of Functions
Functions are often studied and understood as
families, and students should spend time studying
functions within a family, varying parameters to
understand how the parameters affect the graph
of the function and its key features.
Families of Functions
F-IF.7 indicates that the following functions should be in
students’ repertoires:
 Linear and quadratic (show intercepts, maxima,
and minima)
 Square root, cube root, piecewise (step functions
and absolute value functions)
 Polynomial functions (identify zeros when suitable
factorizations available and show end behavior
Families of Functions
F-IF.7 indicates that the following functions should be in
students’ repertoires:
 Exponential and logarithmic functions
 Showing intercepts and end behavior
 Trigonometric functions
 Showing period, midline, and amplitude
 (+) Rational functions
 Identifying zeros and asymptotes when suitable
factorizations available, showing end behavior
F-IF.A.1 Task
 Suppose f is a function.
 If 10=f(−4), give the coordinates of a point on the graph of f.
 If 6 is a solution of the equation f(w)=1, give a point on the graph
of f.
 Commentary:
This task is designed to get at a common student confusion
between the independent and dependent variables. This
confusion often arises in situations like (b), where students are
asked to solve an equation involving a function, and confuse
that operation with evaluating the function.
This task is adapted from Functions Modeling Change: A Preparation for Calculus,
Connally et al., Wiley 2010.
F-IF.A.1 Parking Lot
 A parking lot charges $0.50 for each half hour or fraction
thereof, up to a daily maximum of $10.00. Let C(t) be the cost
in dollars of parking for t minutes.
t (minutes)
 Complete the table
 Sketch a graph of C for 0 ≤ t ≤480.
 Is C a function of t? Explain your reasoning.
 Is t a function of C? Explain your reasoning.
0
15
20
35
75
125
C(t) (dollars)
F-IF.A.2
Let f(t) be the number of people, in millions, who own cell
phones t years after 1990. Explain the meaning of the
following statements.
 f(10)=100.3
 f(a)=20
 f(20)=b
 n=f(t)
from Illustrative Mathematics
Growing Patterns as Functions
 See Handout: Seeing Structure and Developing Rules for
Patterns
 See Fawn Nguyen’s Patterns Poster
 See Patterns Poster for Algebra I
 See Fawn Nguyen’s Visual Patterns Website
Functions in Algebra II
 Interpret functions that arise in applications in terms of a
context (Modeling)
 Extend modeling with functions introduced in Algebra I to
include trigonometric functions and periodicity
Interpreting the graphic model of an influenza epidemic
http://www.illustrativemathematics.org/illustrations/803
Functions in Algebra II
 Analyze functions using different representations
 Focus on using key features to guide selection of
appropriate type of model function
 Extend analysis with function introduced in Algebra I to
include polynomial functions and logarithmic functions
(Read F-IF.7c,d(+), e)
Analyzing the Graphs of Power Functions
http://www.illustrativemathematics.org/illustrations/803
Functions in Algebra II
 Analyze functions using different representations
 Write a function defined by an expression in
different but equivalent forms to reveal and
explain different properties of the function
 Factoring and completing the square (Read
F-IF.8a)
http://www.illustrativemathematics.org/illustratio
ns/803
Functions in Algebra II
 Analyze functions using different representations
 Using properties of exponents
Use the properties of exponents to interpret
expressions for exponential functions. For
example, identify percent rate of change in
functions such as y=(1.02)t, y=(0.97)t, y=(1.01)12t,
y=(1.2)t/10, and classify them as representing
exponential growth or decay.
Functions in Algebra II
 Analyze functions using different representations
 Compare properties of two functions each
represented in a different way (algebraically,
graphically, numerically in tables, or by verbal
descriptions).
For example, given a graph of one quadratic
function and an algebraic expression for
another, say which has the larger maximum.
Functions in Algebra II
 Building Functions
 Continue work started in Algebra I on Building Functions
through adding, subtracting, multiplying, and dividing
functions.
 Include simple radical, rational, and exponential functions
Interpreting and Building an Exponential Model
http://www.illustrativemathematics.org/illustrations/533
Functions in Algebra II
 Building Functions
 Identify the effect of rigid transformations and non-rigid
transformations on the domain and range of a function
 Emphasize common effect of each transformation across
functions types
 Find inverse functions
Functions in Algebra II
Problem in which students need to analyze graphs of
exponential functions (F-IF.7e) and also understand the
effect on the graph of of replacing f(x) by f(x)+k, kf(x), f(kx),
and f(x+k)(F-BF.3)
http://www.illustrativemathematics.org/illustrations/803
Functions in Algebra II
 Linear, Quadratic, and Exponential Models
 Continue work started in Algebra I on
modeling with linear, quadratic, and
exponential functions
 Introduce logarithms as solutions for
exponents.
Bacteria problem
http://www.illustrativemathematics.org/illustrati
ons/370
Functions in Algebra II
Trigonometric Functions
 Understand radian measure of an angle as the length of
the arc on the unit circle subtended by the angle.
 Explain how the unit circle in the coordinate plane
enables the extension of trigonometric functions to all
real numbers, interpreted as radian measures of angles
traversed counterclockwise around the unit circle.
Functions in Algebra II
Trigonometric Functions
 Model periodic phenomena with
trigonometric functions (Read F.TF.5)
Fairly difficult, but accessible, applied trig problem that would
be a good group project:
http://www.illustrativemathematics.org/illustrations/803
Functions in Algebra II
Trigonometric Functions (Honors or 4th Year)
 (+) Understand that restricting a trigonometric
function to a domain on which it is always
increasing or always decreasing allows its
inverse to be constructed.
 (+) Use inverse functions to solve trigonometric
equations that arise in modeling contexts;
evaluate the solutions using technology, and
interpret them in terms of the context. ⋆
Functions in Algebra II
Trigonometric Functions
 Prove and apply trigonometric identities
 Prove the Pythagorean identity sin2(θ)+cos2(θ)=1 and
use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ),
or tan(θ) and the quadrant of the angle.
 (+) Prove the addition and subtraction formulas for sine,
cosine, and tangent and use them to solve problems.
More Functions Tasks
F-IF Pizza Place Promotion
F-BF US Households
F-LE Comparing Exponentials
F-LE Newton's Law of Cooling
F-TF Foxes and Rabbits 2
PARCC Example
Get this from
http://www.parcconline.org/s
amples/mathematics/highschool-functions